Von Neumann's theorem

In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.

Statement of the theorem

Let [math]\displaystyle{ G }[/math] and [math]\displaystyle{ H }[/math] be Hilbert spaces, and let [math]\displaystyle{ T : \operatorname{dom}(T) \subseteq G \to H }[/math] be an unbounded operator from [math]\displaystyle{ G }[/math] into [math]\displaystyle{ H. }[/math] Suppose that [math]\displaystyle{ T }[/math] is a closed operator and that [math]\displaystyle{ T }[/math] is densely defined, that is, [math]\displaystyle{ \operatorname{dom}(T) }[/math] is dense in [math]\displaystyle{ G. }[/math] Let [math]\displaystyle{ T^* : \operatorname{dom}\left(T^*\right) \subseteq H \to G }[/math] denote the adjoint of [math]\displaystyle{ T. }[/math] Then [math]\displaystyle{ T^* T }[/math] is also densely defined, and it is self-adjoint. That is, [math]\displaystyle{ \left(T^* T\right)^* = T^* T }[/math] and the operators on the right- and left-hand sides have the same dense domain in [math]\displaystyle{ G. }[/math]